Lyapunov-Metzler Inequalities Research Articles

Novel Practical Stability Conditions for Discrete-Time Switched Affine Systems

This technical note proposes novel state feedback stability conditions based on the so called Lyapunov–Metzler inequalities for discrete-time switched affine systems, assuring practical stability of a desired equilibrium point. These conditions are obtained taking into account the volume minimization of an ellipsoidal set containing the nonconvex set of attraction to where the state trajectories are globally attracted. Compared with other recent results, the present ones are based on less conservative conditions for the existence of an attraction set. Moreover, it is not required that the equilibrium point be inside a predetermined set of attainable ones, which enlarges the range of applicability of the present technique. To the best of the authors’ knowledge, it is the first time that the Lyapunov–Metzler inequalities are adopted in the context of switched affine systems. As a second step, the associated nonconvex invariant set is provided. Academical examples illustrate the method and compare the theoretical results with recent ones available in the literature.

ℋ2 static output feedback switching function design based on a time-varying Lyapunov function approach

This paper treats the control design of an output-dependent switching function for discrete-time switched linear systems, without considering the synthesis of any additional dynamic structure. The conditions are based on a convex, but time-varying, Lyapunov function and are expressed in terms of Linear Matrix Inequalities (LMIs). The state feedback switching function design can be obtained as a particular case and is simpler to handle from theoretical and numerical viewpoints compared to other methods available in the literature, as for instance, those based on Lyapunov–Metzler inequalities. The novelty of the proposed control methodology is the adoption of a time-varying Lyapunov function, which together with the presence of a key matrix, allow us to treat the static output feedback control design, taking into account a guaranteed performance cost. Academic examples are used for comparison and illustration. The proposed control technique is experimentally validated in an active suspension system representing a quarter-vehicle.

Stabilization of Positive Switched Linear Time-Varying Systems With Delay Under State-Dependent Switching

In this paper, we focus on the stabilization of positive switched linear time-varying systems with delay. By constructing Lyapunov-Metzler inequalities, state-dependent switching signals have been designed such that the system is asymptotically stable, even if each subsystem is not asymptotically stable. Both the continuous-time and the discrete-time positive switched linear time-varying systems have been taken into consideration. Numerical examples are also given to verify the validity of our results.

Stability Analysis of Switched Linear Systems with Uncertainty and Delays

Combining the common quadratic Lyapunov functional approach and free-weighting matrix approach, this paper is devoted to the stability analysis of continuous-time switched linear systems (CTSLS) with uncertainty and time-delays. A particular class of matrix inequalities, the so-called Lyapunov–Metzler inequalities are introduced for the CTSLS to investigate the stability and performance in the presence of the uncertainty and delays. We provide sufficient conditions in terms of the Linear Matrix Inequality criterions to guarantee delay-dependent asymptotically stability under the uncertainty of the CTSLS. The combination of switching rule and switching output feedback controllers which will be designed to stabilize the CTSLS and satisfy a prespecified $${\mathcal{L}}_{2}$$ gain performance. A example used jitterbug tools is provided to illustrate the effectiveness of the proposed method.

Switched Optimized Robust Control of Uncertain LPV Systems Subject to Structural Faults

Abstract This paper proposes improved techniques for the control of continuous-time, switched uncertain linear LPV systems with decay rate performance index and optimization of the controller’s norm. These techniques are of closed loop nature (trajectory dependent) and are designed using the solution of Lyapunov-Metzler inequalities from which the stability condition is expressed. The performance index used in the design procedure is the decay rate that aims to reduce the system’s settling time. The other important tool proposed is the optimization of the controller’s norm that aims to reduce the gains and facilitate its application. In addition to this, more relaxed conditions than those existing in the literature to obtain switched controllers are proposed, which will enable better global performance. Practical implementation of the improved techniques were made in a Quanser’s Active Mass Damper - One Floor (AMD-1) laboratory prototype, subject to a structural failure, addressed as a polytopic time-varying uncertainty.

On Lyapunov-Metzler Inequalities and S-Procedure Characterizations for the Stabilization of Switched Linear Systems

In this note we present connections between two celebrated tools for the design of stabilising switching laws for continuous-time and discrete-time switched linear systems, namely Lyapunov-Metzler inequalities and S-procedure.

Stabilization of discrete-time switched linear systems: Lyapunov-Metzler inequalities versus S-procedure characterizations

In this paper we study connections between Lyapunov-Metzler inequalities and S-procedure characterizations in the context of stabilizing discrete-time switched linear systems using min-switching strategies. We propose two generalized versions of S-procedure characterization along the lines of the generalized versions of Lyapunov-Metzler inequalities recently proposed in the literature. It is shown that the existence of a solution to the generalized version(s) of Lyapunov-Metzler inequalities is equivalent to the existence of a solution to the generalized version(s) of S-procedure characterization with a restricted choice of the scalar quantities involved in the latter. This recovers some of our earlier works on the classical Lyapunov-Metzler inequalities as a special case. We also highlight and discuss an open question of whether the generalized versions of S-procedure characterization are strictly less conservative than the generalized versions of Lyapunov-Metzler inequalities, which in turn are equivalent to periodic stabilizability as was recently shown.

Quality Assessment of Lyapunov-Metzler Methods for Discrete-Time Switched Control

This paper addresses the quality of the solutions provided by Lyapunov-Metzler inequalities to Discrete Switched System Control. It proposes a method based on the value iteration algorithm, which is able to efficiently compute an approximately optimal solution to the dynamic programming formulation arising from these systems. A large number of numerical simulations are performed in order to evaluate the results.

From adaptive control to variable structure systems – seeking harmony

From adaptive control to variable structure systems – seeking harmony

Formula omitted] and [formula omitted] control of time-varying delay switched linear systems with application to sampled-data control

This paper deals with switched linear systems subject to time-varying delay. The main goal is to design state and output feedback switching strategies preserving closed-loop stability and a guaranteed H2 or H∞ performance. The switching strategies are based on a generalization of a recent extended version of the small gain theorem and do not require any assumption on the continuity of the delay and its time-variation rate. The key point to obtain the design conditions is the adoption of an equivalent switched linear system where the time-varying delay is modeled as a norm-bounded perturbation. Moreover, with this approach, it is possible to deal with sampled-data control systems. All conditions are formulated in terms of Lyapunov–Metzler inequalities, which allow the maximization of an upper bound on the time-delay preserving stability and guaranteed performance. Numerical examples are discussed in order to illustrate the effectiveness of the design approach.

Robust switched control based on strictly positive real systems and variable structure control techniques

SummaryThis paper is concerned with the design of variable structure controllers for uncertain switched linear plants. The proposed method is based on Lyapunov–Metzler inequalities and on properties of strictly positive real (SPR) systems, with the advantage that it can be applied in control of uncertain switched linear system. The first contribution shows that there exists a solution of the SPR synthesis for a class of linear uncertain switched systems using switched static output feedback, if and only if there exists a solution of this SPR synthesis with switched static state feedback. This result is a generalization of known conditions regarding this problem, for linear time‐invariant and known (without uncertainties) plants, and offers a possible procedure for output feedback SPR synthesis: first solve the state feedback SPR synthesis and then, using the obtained solution, find a solution for the output feedback SPR synthesis. Next are presented necessary and sufficient (sufficient) conditions for the SPR synthesis for a class of linear uncertain switched systems using switched static output feedback, for plants with the same number of inputs and outputs (with the number of outputs greater than the number of inputs). Two examples illustrate the effectiveness of the robust control system, including applications of the proposed methods, considering switched static state feedback and switched static output feedback, in the design of switching control strategies for active suspensions systems in road vehicles. Performance indices such as decay rate, guaranteed cost, and restriction in the norm of controller gains are considered in these designs. Copyright © 2016 John Wiley & Sons, Ltd.

On the Stabilizability of Discrete-Time Switched Linear Systems: Novel Conditions and Comparisons

In this paper we deal with the stabilizability property for discrete-time switched linear systems. A recent necessary and sufficient characterization of stabilizability, based on set theory, is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with computational affordability. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity.

Quadratic and H∞ switching control for discrete-time linear systems with multiplicative noises

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H∞ performance costs are bounded above by a constant value. These results are illustrated with some numerical examples.

Consensus of second-order multi-agent systems with nonlinear dynamics and switching topology

This paper is concerned with the consensus problem in second-order multi-agent systems with nonlinear dynamics and switching topology. The switching rule is assumed to switch among a finite number of undirected graphs and be trajectory dependent. By using Lyapunov–Metzler inequalities, some easily checkable conditions are obtained to guarantee that consensus can be achieved by a state switching rule. Two examples are presented to illustrate the effectiveness of the obtained results.

Robust [formula omitted] switched filter design for discrete-time polytopic linear parameter-varying systems

This paper deals with a robust H2 filter design problem for discrete-time polytopic linear parameter-varying systems. The novelty is to design a set of full order filters and a switching rule to orchestrate them in order to assure an H2 performance index of the estimation error. The switched linear filter does not require the online measurement of the uncertain time-varying parameter which, in a great part of practical applications, is not available. The conditions are based on Lyapunov–Metzler inequalities with a special subclass of Metzler matrices. The proposed switched filter outperforms the ones available in the literature to date as far as the H2 norm of the estimation error is taken into account. The main theoretical implications of the switched filtering methodology in the framework of linear time-invariant uncertain systems and linear parameter-varying systems are discussed and illustrated by academical examples.

Stability of Discrete-Time Switched Linear Systems with Saturation Arithmetic

This paper addresses a strategy for the stability of discrete-time switched linear systems with saturation arithmetic. It is of closed-loop nature and is designed from the solution of what we called Lyapunov-Metzler inequalities from which the stability condition is expressed. The theoretical results are illustrated by means of examples.

Discussion on: “Suboptimal switching control consistency analysis for discrete-time switched linear systems”

The past 20 years has seen a huge number of papers produced in the area of switched and hybrid systems. The principal focus of these papers has been to produce conditions under which stability can be guaranteed, either in the case of arbitrary switching, or in the case of constrained switching, where one constrains the switching through state constraints, or through limits on the rate of switching. Some important topics have, however, rather disappointingly, been the subject of far less study. These include the discretization of switched systems, the design of systems where constraints are fixed rather than a design variable, and the design of switched systems with performance guarantees (to name but a few). A major contribution of the paper under discussion is that it falls into the latter category, and provides results that are likely to be of interest, not only to the academic community, but also to practicing engineers. Following similar work in the context of continuous-time switched linear systems [1], the authors introduce the concept of consistency for the discrete-time counterpart. More specifically, a given switching strategy is consistent if it improves the performance of the closed loop system when compared with any of the isolated subsystems. The performance of the controlled system is evaluated in terms of input–output H2 and H∞ indexes, which are defined in such a way that whenever the switching policy is constant, then the indices coincide with the conventional definition for the single subsystem. Section 3 constitutes the main part of the paper. The authors present two theorems that guarantee the existence of a consistent switching function; the first based on conditions on the Hi matrices, and the second based on even other conditions on the class of switching systems considered. In both cases the use of Lyapunov–Metzler inequalities is heavily employed. Note in the discussion following both of these theorems, the investigation of strict consistency involves the formulation of a non-convex search that would almost certainly be

Min-switching local stabilization for discrete-time switching systems with nonlinear modes

This paper deals with the discrete-time switched Lur’e problem in finite domain. The aim is to provide a stabilization inside an estimate of the origin’s basin of attraction, as large as possible, via a suitable switching rule. The design of this switching rule is based on the min-switching policy and can be induced by sufficient conditions given by Lyapunov–Metzler inequalities. Nevertheless instead of intuitively considering a switched quadratic Lyapunov function for this approach, a suitable switched Lyapunov function including the modal nonlinearities is proposed. The assumptions required to characterize the nonlinearities are only mode-dependent sector conditions, without constraints related to the slope of the nonlinearities. An optimization problem is provided to allow the maximization of the size of the basin of attraction estimate–which may be composed of disconnected sets–under the stabilization conditions. A numerical example illustrates the efficiency of our approach and emphasizes the specificities of our tools.

Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach*

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but not a necessary and sufficient one. The main contribution is a computation-oriented necessary and sufficient condition for stabilizability based on set-theory. Based on such a condition, an algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided. The generic algorithm is based on the invariance of unions of compact, convex sets containing the origin and is applied to numerical examples using ellipsoids and polytopes. It will be shown in particular that no assumption is made on the existence of a Schur convex combination of the matrices, assumption on which the Lyapunov-Metzler inequalities approach is based. Further discussions with respect to the literature and concluding remarks are also proposed.

Analysis and control of switched linear systems via modified Lyapunov–Metzler inequalities

SUMMARYThis paper addresses analysis and switching control problems of continuous/discrete‐time switched linear systems. A particular class of matrix inequalities, the so‐called Lyapunov–Metzler inequalities, will be modified to provide conditions for stability analysis and output feedback control synthesis under a relaxed min‐switching logic. The switching rule combined with switching output feedback controllers will be designed to stabilize the switched system and satisfy a prespecified gain performance. The proposed analysis and switching control approach could refrain frequent switches commonly observed in min‐switching based designs. The effectiveness of the proposed approach will be illustrated through numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.